Multiscale methods and analysis for the highly oscillatory nonlinear Klein-Gordon equation

Date: 2021-07-15

Time: 09:00 - 09:45

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Weizhu Bao


In this talk, I begin with the nonlinear Klein-Gordon equation (NKGE) under two important parameter regimes, i.e. one is nonrelativisitic regime and the other is long-time dynamics with weak nonlinearity or small initial data, while the NKGE is highly oscillatory. I first review our recent works on numerical methods and analysis for solving the NKGE in the nonrelativistic regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. Then we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the NKGE. Rigorious error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the NKGE in the nonrelativistic regime. In order to design a multiscale method for the NKGE, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schrodinger equation perturbed with a wave operator. Based on a large-small amplitude wave decompostion to the solution of the NKGE, a multiscale method is presented for discretizing the NKGE in the nonrelativistic regime. Rigorous error estimates show that this multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the NKGE in the nonrelativistic regime. Finally, I discuss issues related error bounds of different numerical methods for the long-time dynamics of NKGE with weak nonlinearity and applications to several highly oscillatory dispersive partial differential equations.